For long chains (\(N\gg 10\) ) we may approximate our discrete chain with a continuous chain parameterized with the Rouse curvilinear coordinate \(n\)  [1]}.
Furthermore, as we only require the ensemble averaged positions we may average our equation of motion (eliminating the Brownian force) resulting in:
\(\zeta \frac{\partial \langle x_n \rangle }{\partial t} = \frac{2kT}{b^2}\frac{\partial ^2 \langle x_n \rangle }{\partial n^2};\) 
\(\langle x_0 \rangle = -v_pt;\) 
\(\frac{\partial \langle x_N \rangle }{\partial n} = \bar{L}/N.\) 
